How to prove trigonometric identities — strategy and common tricks

Question

Prove that: $\frac{\sin\theta}{1 + \cos\theta} + \frac{1 + \cos\theta}{\sin\theta} = 2\csc\theta$

Show the strategy for approaching such proofs systematically.

(CBSE Class 10/11 pattern)

Solution — Step by Step

flowchart TD
    A["Trig Identity Proof"] --> B{"Which side\nis more complex?"}
    B -->|"LHS is complex"| C["Simplify LHS\nto match RHS"]
    B -->|"RHS is complex"| D["Simplify RHS\nto match LHS"]
    B -->|"Both complex"| E["Simplify both\nto a common form"]
    C --> F["Key tricks:\n1. Convert to sin/cos\n2. Take LCM\n3. Use identities\n4. Rationalise"]

Here, LHS has two fractions — we will simplify LHS to get RHS.

\text{LHS} = \frac{\sin\theta}{1 + \cos\theta} + \frac{1 + \cos\theta}{\sin\theta}

LCM of denominators is $\sin\theta(1 + \cos\theta)$ :

= \frac{\sin^2\theta + (1 + \cos\theta)^2}{\sin\theta(1 + \cos\theta)}

\sin^2\theta + 1 + 2\cos\theta + \cos^2\theta

Using $\sin^2\theta + \cos^2\theta = 1$ :

= 1 + 1 + 2\cos\theta = 2 + 2\cos\theta = 2(1 + \cos\theta)

\text{LHS} = \frac{2(1 + \cos\theta)}{\sin\theta(1 + \cos\theta)} = \frac{2}{\sin\theta} = 2\csc\theta = \text{RHS}

Hence proved.

Why This Works

Most trig identity proofs reduce to algebraic manipulation once everything is in terms of $\sin$ and $\cos$ . The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ is the workhorse — it appears in nearly every proof. The strategy of taking LCM converts addition of fractions into a single fraction, making simplification easier.

The proof flows because we treated the trig expressions as algebraic fractions and applied standard fraction operations. No special trig insight is needed beyond knowing the fundamental identities.

Alternative Method — Rationalisation Approach

Multiply the first fraction by $\frac{1 - \cos\theta}{1 - \cos\theta}$ :

\frac{\sin\theta(1-\cos\theta)}{(1+\cos\theta)(1-\cos\theta)} = \frac{\sin\theta(1-\cos\theta)}{\sin^2\theta} = \frac{1-\cos\theta}{\sin\theta}

Then: $\frac{1-\cos\theta}{\sin\theta} + \frac{1+\cos\theta}{\sin\theta} = \frac{2}{\sin\theta} = 2\csc\theta$ .

This is often faster when you spot the conjugate pattern.

Five golden tricks for trig proofs: (1) Convert everything to sin and cos, (2) Use $\sin^2 + \cos^2 = 1$ and its rearrangements, (3) Take LCM for fraction sums, (4) Rationalise using conjugates like $(1-\cos\theta)$ , (5) Factor expressions like $a^2 - b^2 = (a+b)(a-b)$ . These five tricks handle 90% of CBSE trig proofs.

Common Mistake

The cardinal rule of trig proofs: never work on both sides simultaneously and meet in the middle. This is a common shortcut students take, but it is logically invalid — you would be assuming what you are trying to prove. Always start from one side (usually the more complex one) and arrive at the other. Examiners specifically deduct marks for “both-sides” proofs.

Question

Solution — Step by Step

Why This Works

Alternative Method — Rationalisation Approach

Common Mistake

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